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Promotion of cooperation by aspiration-induced migration
Physica A 390 (2011) 77–82
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Physica A journal homepage: www.elsevier.com/locate/physa
Promotion of cooperation by aspiration-induced migration Ying-Ting Lin a , Han-Xin Yang a,∗ , Zhi-Xi Wu b , Bing-Hong Wang a,c a
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
b
Department of Physics, Umeå University, 90187 Umeå, Sweden
c
The Research Center for Complex System Science, University of Shanghai for Science and Technology and Shanghai Academy of System Science, Shanghai, 200093, China
article
info
Article history: Received 25 May 2010 Received in revised form 4 July 2010 Available online 6 August 2010 Keywords: Prisoner’s dilemma game Cooperation Migration Aspiration
abstract In this paper, we study an aspiration-induced migration model, in which each individual plays the prisoner’s dilemma game with those being within a circle of radius r centered on himself/herself. An individual will migrate to a randomly chosen place with the velocity v if his/her payoff is below the aspiration level. We report that cooperative behavior is favored when the aspiration level and interaction radius are moderate, and the migration velocity is slow. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Cooperation is fundamental to biological and social systems. So far, evolutionary game theory [1,2] has provided a powerful framework for understanding the emergence of cooperation. Many important mechanisms have been considered for studying cooperative behavior, such as costly punishment [3,4], reputation [5,6], social diversity [7–9], coevolution of strategy and structure [10–14], mortality selection [15] and so on. Migration is ubiquitous in the animal world and human society. For example, millions of animals migrate in the savannas of Africa every year and thousands of people travel among different countries every day. Recently, there has been much interest in studying the effect of migration on the evolution of cooperation [16–23]. In previous studies, migration can be in a random-walk way or driven by some special mechanisms. Vainstein et al. [18] studied the case in which individuals are located on the sites of a two-dimensional regular lattice and each individual makes an attempt to jump to a nearest neighboring site chosen randomly, provided the site is empty, with a probability. They found that such movement can maintain and even enhance cooperation compared to the never-move case. Helbing and Yu [19] proposed a success-driven migration in which individuals will move to sites with the highest estimated payoffs. It was found that such migration results in the outbreak of cooperation in a noisy environment. Meloni et al. [21] considered the case in which individuals are situated on a two-dimensional plane and each individual moves to a randomly chosen position with the speed v . Their results showed that cooperation can survive provided that both the temptation to defect and the speed are not too high. Jiang et al. proposed [22] an adaptive migration strategy in which individuals are located on a square lattice and an individual moves to an empty site with the probability proportional to the number of defectors in its neighborhood. They found that adaptive migration can induce an outbreak of cooperation from an environment dominated by defectors. It is noted that in some real-life situations, individuals have to migrate when their current places are not suitable for living. For example, animals will migrate to other places if they cannot find enough food in their current habitats and many
∗
Corresponding author. E-mail addresses: [email protected] (H.-X. Yang), [email protected] (B.-H. Wang).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.07.034
78
Y.-T. Lin et al. / Physica A 390 (2011) 77–82
islanders have to leave their homeland as the sea level rises. In Ref. [23], Yang et al. proposed an aspiration-induced migration model. In their model, individuals are situated on a square lattice and gain payoffs by playing against individuals sitting on four neighboring nodes. An individual will move to a randomly chosen empty site within its four neighboring sites if its payoff is lower than the aspiration level. Here the aspiration level can also be understood as the minimum living standard. In this paper, we also study the effect of aspiration-induced migration on cooperation but consider the less-constrained case in which individuals are located on a two-dimensional plane. Two individuals play with each other if the distance between them is less than r. Individuals will move to other places with the velocity v if their payoffs are below the aspiration level. As we will show, the highest cooperation level is achieved when the aspiration level and interaction radius are moderate, and the migration velocity is slow. The paper is organized as follows. In the next section, the aspiration-induced migration on a two-dimensional plane is introduced. The simulation results and discussions are given in Section 3 and the paper is concluded by the last section. 2. The model Our study is carried out for the prisoner’s dilemma game (PDG) [24]. In a PDG played by two players, each chooses one of two strategies, cooperation or defection. They both receive payoff R upon mutual cooperation and P upon mutual defection. If one defects while the other cooperates, cooperator receives S while defector gets T . The ranking of the four payoff values is: T > R > P > S. Thus in a single round of the PDG it is best to defect regardless of the opponent’s decision. The PDG has attracted much attention in theoretical and experimental studies of cooperative behavior. Following common practice [25], we set T = b (b > 1), R = 1, and P = S = 0, where b represents the temptation to defect. As the pioneering work, Nowak and May included spatial structure in the PDG [25], such that individuals are constrained to play only with their immediate neighbors. There is much current interest in studying evolutionary games in structured populations and on networks [25–54]. Individuals’ neighborhoods are fixed in a statically structured population. However, in the case of migration, individuals can change their neighborhoods. Following previous study [21], we assume N individuals are located on a continuous square plane of linear size L with periodic boundary conditions. Initially, an equal percentage of strategies (cooperators or defectors) is randomly distributed among the population. The movement and game dynamics might in general be correlated [55], and the influence of individuals’ movement on the performance of the PDG dynamics might depend on the ratio between their corresponding time scales. Here, we consider the situation in which both movement and evolutionary dynamics have the same time scale. At each time step, the neighborhood of a given individual i is made up by all the individuals being within a circle of radius r that centered at individual i. Once individuals have played the PDG with all their neighbors, they accumulate payoffs obtained in each game. All strategies are updated synchronously by following the finite population analogue of the replicator dynamics [27]. When an individual i with a payoff Pi is selected for update, a neighbor j (with a payoff Pj ) is drawn at random among all ki neighbors. If Pi ≥ Pj , no update occurs. If Pi < Pj , i will adopt j’s strategy with a probability given by (Pj − Pi )/bk> , where k> is the larger between ki and kj . Whenever updating strategy, an individual also decides whether to stay in or leave his/her current position. If his/her payoff reaches or exceeds his/her aspiration level, he/she does not move. Otherwise, he/she updates his/her positions according to: xi+1 (t + 1) = xi (t ) + vi (t ),
(1)
θi+1 (t + 1) = ηi ,
(2)
where xi is the position of the individual i in the plane at time t, the velocity of individual vi (t ) is constructed to have an absolute value v and a direction given by the angle θi (t ), and ηi is a random number chosen with uniform probability in the interval [−π , π]. To avoid isolated cases, we assume that an isolated individual determinately moves. Following previous study [23,56], the aspiration level Pia for an individual i is defined as Pia = ki U, where ki is the degree of individual i (in this paper, individual i’s degree is the number of individuals being within a circle that centered at individual i) and U is a control parameter (U is the same for all individuals). This definition is based on the following consideration: Maintaining a social contact is usually costly [57]. We assume for simplicity that an individual pays U cost to maintain a link with one of its neighbors, and the aspiration level for an individual is defined as the total cost for maintaining social links with all its neighbors. Thus it is reasonable to assume the aspiration level to be proportional to the degree of individuals. For U ≤ 0, the minimal payoff of individuals is higher than the aspiration level, individuals do not move (except isolated individuals). For U > b, the aspiration level is higher than the maximal payoff of individuals, all individuals move. 3. Results and discussions Fig. 1 shows the fraction of cooperators ρc as a function of the aspiration level U for different values of the temptation to defect b when the migration velocity v = 0.2. One can find that for a fixed value of b, there exists a moderate aspiration level U, resulting in the highest cooperation level. Fig. 2 depicts the results obtained for a wider range of model parameters (U and b). As shown in Fig. 2, there exists an intermediate aspiration level, resulting in the highest cooperation level for different values of b. Additionally, the maximal cooperation level at the optimal intermediate aspiration level decreases with the increase of b.
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Fig. 1. (Color online) The fraction of cooperators ρc as a function of aspiration level U for different values of the temptation to defect b. The population size N = 1024, the interaction radius r = 1, the migration velocity v = 0.2 and the linear size of square plane L = 25. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
Fig. 2. (Color online) The color code shows ρc as a function of U and b together. Other parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators’ results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
It is known that in spatial games, cooperators can survive by forming clusters [58,59], thus reducing exploitation by defectors. For low aspiration levels, most individuals do not move, thus cooperator clusters and defector clusters keep relatively steady, restraining the pervasion of cooperators among population. On the contrary, for high aspiration levels, most individuals move. Due to the frequent change of neighbors, cooperators cannot form steady clusters to resist invasion of defectors. As a result, cooperators are extinct and defectors dominate the whole population. For moderate aspiration levels, cooperators are able to form steady clusters since the benefits of mutual cooperation are high, whereas defector clusters are unstable because the payoffs of mutual defection are low. From Fig. 3, one can see that for moderate aspiration levels, the number of mobile defectors is much more than that of mobile cooperators during the process of evolution. Furthermore, one can see that the number of mobile cooperators decreases as time evolves, while the number of mobile defectors increases firstly and then decreases as time evolves. Once a mobile defector approaches the boundary of a cooperator cluster, he/she has a large probability to change to a cooperator since cooperator clusters usually obtain a high payoff. Compared with the static case in which cooperator clusters and defector clusters are almost changeless, a moderate aspiration level leads to the expansion of cooperator clusters and the collapse of defector clusters. So, cooperation is promoted when the aspiration level is moderate. To confirm the above analysis, we study the fraction of cooperators ρc as a function of time for different aspiration levels. As shown in Fig. 4, for a low aspiration level (U = 0), ρc reaches a steady value, indicating that cooperator clusters
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Fig. 3. (Color online) The number of mobile cooperators nc and mobile defectors nd as a function of time step t. The population size N = 512, the interaction radius r = 1.3, the migration velocity v = 0.3, the temptation to defect b = 1.1, the linear size of square plane L = 25 and the aspiration level U = 0.5.
Fig. 4. (Color online) The fraction of cooperators ρc as a function of time step t for different values of U. Other model parameters are the same as those in Fig. 3.
and defector clusters coexist and remain stable in the equilibrium state. For a moderate aspiration level (U = 0.5), ρc decreases in the beginning and then increases to unity as time evolves. This is because cooperators are initially dispersive and they are vulnerable to the attacks of defectors, resulting in the decrease of ρc at the beginning. After cooperators gather together to form clusters, cooperator clusters will expand continually when the aspiration level is moderate, thus ρc increases. For a high aspiration level (U = 1.1), ρc decreases to zero, indicating that cooperators are extinct in the end. We provide an intuitionistic observation for the evolution of cooperation in supplementary movies S1–S3 (available online at doi:10.1016/j.physa.2010.07.034). Movie S1, S2 and S3 corresponds to U = 0, 0.5 and 1.1 respectively. In all movies, blue spots represent cooperators and red spots denote defectors. The migration velocity v also affects the evolution of cooperation. Low migration velocity favors cooperation while high migration velocity promotes the spread of defection. As shown in Fig. 5(a), the fraction of cooperators ρc decreases as the migration velocity v increases for different values of the aspiration level U. Fig. 5(b) shows that ρc as a function of U for different values of v . One can see that, for v = 0.005, 0.1 and 0.2, the cooperation level is highest when U is moderate. Additionally, the maximal cooperation level at moderate aspiration level decreases with the increase of v . In particular, for high migration velocity (v = 0.5), cooperators can survive when the aspiration level U = 0 (note that U = 0 corresponds to the static case in which individuals do not move), while defectors dominate the whole system for U > 0. For low migration velocity, a mobile defector slowly approaches the boundary of the cooperator cluster and he/she is likely transfer to cooperator due to the influence of cooperators locating inside cooperator clusters. However, for high migration velocity,
Y.-T. Lin et al. / Physica A 390 (2011) 77–82
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b 0.005 0.1 0.2 0.5
9
–2
–1
0
Fig. 5. (Color online) (a) The fraction of cooperators ρc as a function of the velocity v for different values of U. (b) ρc as a function of the aspiration level U for different values of the velocity v . The temptation to defect b = 1.25, other model parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps (for v = 0.005, the equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 106 time steps). Each data point is an average over 500 different realizations.
Fig. 6. (Color online) The fraction of cooperators ρc as a function of the interaction radius r for different values of U. The temptation to defect b = 1.2, other model parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
a mobile defector will rapidly enter the core of acooperator cluster, thus gaining high payoff and resulting in the collapse of the cooperator clusters. The neighborhood size plays an important role in cooperation [60]. Fig. 6 studies the fraction of cooperators ρc as a function of the interaction radius r for different aspiration levels. One can see that there exists a moderate value of r, leading to the highest cooperation level. For a small interaction radius r, individuals have few neighbors and cooperators cannot form clusters to resist the invasion of defectors. For a large interaction radius r, individuals’ neighborhoods resemble a wellmixed population in which more or less everybody interacts with everybody, so that the low cooperation level is inevitable. 4. Conclusion In summary, we have studied the effect of aspiration-induced migration on the evolution of cooperation. Individuals are located on a square plane and gain payoffs by playing the prisoner’s dilemma game with those within a circle of radius r. Individuals will leave their current locations if their payoffs are lower than the aspiration level. We have found that there exists a moderate aspiration level, leading to the highest cooperation level. Additionally, cooperation is enhanced if migrants move with low velocity but inhibited if they move with high velocity. For a moderate aspiration level plus a low migration velocity, cooperator clusters outspread and defector clusters disintegrate, resulting in the prevalence of cooperation amongst the population. Moreover, we have reported that a moderate interaction radius can best promote cooperation. Our study may be of practical significance and help to explain why cooperation is ubiquitous in nature. Furthermore, our work may encourage in-depth studies on the role of migration in evolutionary game theory.
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Acknowledgements We thank Hisashi Ohtsuki for helpful discussions. This work is funded by the National Basic Research Program of China (973 Program No. 2006CB705500), the National Natural Science Foundation of China (Grant Nos. 10975126, 10635040), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20093402110032). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Promotion of cooperation by aspiration-induced migration Ying-Ting Lin a , Han-Xin Yang a,∗ , Zhi-Xi Wu b , Bing-Hong Wang a,c a
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
b
Department of Physics, Umeå University, 90187 Umeå, Sweden
c
The Research Center for Complex System Science, University of Shanghai for Science and Technology and Shanghai Academy of System Science, Shanghai, 200093, China
article
info
Article history: Received 25 May 2010 Received in revised form 4 July 2010 Available online 6 August 2010 Keywords: Prisoner’s dilemma game Cooperation Migration Aspiration
abstract In this paper, we study an aspiration-induced migration model, in which each individual plays the prisoner’s dilemma game with those being within a circle of radius r centered on himself/herself. An individual will migrate to a randomly chosen place with the velocity v if his/her payoff is below the aspiration level. We report that cooperative behavior is favored when the aspiration level and interaction radius are moderate, and the migration velocity is slow. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Cooperation is fundamental to biological and social systems. So far, evolutionary game theory [1,2] has provided a powerful framework for understanding the emergence of cooperation. Many important mechanisms have been considered for studying cooperative behavior, such as costly punishment [3,4], reputation [5,6], social diversity [7–9], coevolution of strategy and structure [10–14], mortality selection [15] and so on. Migration is ubiquitous in the animal world and human society. For example, millions of animals migrate in the savannas of Africa every year and thousands of people travel among different countries every day. Recently, there has been much interest in studying the effect of migration on the evolution of cooperation [16–23]. In previous studies, migration can be in a random-walk way or driven by some special mechanisms. Vainstein et al. [18] studied the case in which individuals are located on the sites of a two-dimensional regular lattice and each individual makes an attempt to jump to a nearest neighboring site chosen randomly, provided the site is empty, with a probability. They found that such movement can maintain and even enhance cooperation compared to the never-move case. Helbing and Yu [19] proposed a success-driven migration in which individuals will move to sites with the highest estimated payoffs. It was found that such migration results in the outbreak of cooperation in a noisy environment. Meloni et al. [21] considered the case in which individuals are situated on a two-dimensional plane and each individual moves to a randomly chosen position with the speed v . Their results showed that cooperation can survive provided that both the temptation to defect and the speed are not too high. Jiang et al. proposed [22] an adaptive migration strategy in which individuals are located on a square lattice and an individual moves to an empty site with the probability proportional to the number of defectors in its neighborhood. They found that adaptive migration can induce an outbreak of cooperation from an environment dominated by defectors. It is noted that in some real-life situations, individuals have to migrate when their current places are not suitable for living. For example, animals will migrate to other places if they cannot find enough food in their current habitats and many
∗
Corresponding author. E-mail addresses: [email protected] (H.-X. Yang), [email protected] (B.-H. Wang).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.07.034
78
Y.-T. Lin et al. / Physica A 390 (2011) 77–82
islanders have to leave their homeland as the sea level rises. In Ref. [23], Yang et al. proposed an aspiration-induced migration model. In their model, individuals are situated on a square lattice and gain payoffs by playing against individuals sitting on four neighboring nodes. An individual will move to a randomly chosen empty site within its four neighboring sites if its payoff is lower than the aspiration level. Here the aspiration level can also be understood as the minimum living standard. In this paper, we also study the effect of aspiration-induced migration on cooperation but consider the less-constrained case in which individuals are located on a two-dimensional plane. Two individuals play with each other if the distance between them is less than r. Individuals will move to other places with the velocity v if their payoffs are below the aspiration level. As we will show, the highest cooperation level is achieved when the aspiration level and interaction radius are moderate, and the migration velocity is slow. The paper is organized as follows. In the next section, the aspiration-induced migration on a two-dimensional plane is introduced. The simulation results and discussions are given in Section 3 and the paper is concluded by the last section. 2. The model Our study is carried out for the prisoner’s dilemma game (PDG) [24]. In a PDG played by two players, each chooses one of two strategies, cooperation or defection. They both receive payoff R upon mutual cooperation and P upon mutual defection. If one defects while the other cooperates, cooperator receives S while defector gets T . The ranking of the four payoff values is: T > R > P > S. Thus in a single round of the PDG it is best to defect regardless of the opponent’s decision. The PDG has attracted much attention in theoretical and experimental studies of cooperative behavior. Following common practice [25], we set T = b (b > 1), R = 1, and P = S = 0, where b represents the temptation to defect. As the pioneering work, Nowak and May included spatial structure in the PDG [25], such that individuals are constrained to play only with their immediate neighbors. There is much current interest in studying evolutionary games in structured populations and on networks [25–54]. Individuals’ neighborhoods are fixed in a statically structured population. However, in the case of migration, individuals can change their neighborhoods. Following previous study [21], we assume N individuals are located on a continuous square plane of linear size L with periodic boundary conditions. Initially, an equal percentage of strategies (cooperators or defectors) is randomly distributed among the population. The movement and game dynamics might in general be correlated [55], and the influence of individuals’ movement on the performance of the PDG dynamics might depend on the ratio between their corresponding time scales. Here, we consider the situation in which both movement and evolutionary dynamics have the same time scale. At each time step, the neighborhood of a given individual i is made up by all the individuals being within a circle of radius r that centered at individual i. Once individuals have played the PDG with all their neighbors, they accumulate payoffs obtained in each game. All strategies are updated synchronously by following the finite population analogue of the replicator dynamics [27]. When an individual i with a payoff Pi is selected for update, a neighbor j (with a payoff Pj ) is drawn at random among all ki neighbors. If Pi ≥ Pj , no update occurs. If Pi < Pj , i will adopt j’s strategy with a probability given by (Pj − Pi )/bk> , where k> is the larger between ki and kj . Whenever updating strategy, an individual also decides whether to stay in or leave his/her current position. If his/her payoff reaches or exceeds his/her aspiration level, he/she does not move. Otherwise, he/she updates his/her positions according to: xi+1 (t + 1) = xi (t ) + vi (t ),
(1)
θi+1 (t + 1) = ηi ,
(2)
where xi is the position of the individual i in the plane at time t, the velocity of individual vi (t ) is constructed to have an absolute value v and a direction given by the angle θi (t ), and ηi is a random number chosen with uniform probability in the interval [−π , π]. To avoid isolated cases, we assume that an isolated individual determinately moves. Following previous study [23,56], the aspiration level Pia for an individual i is defined as Pia = ki U, where ki is the degree of individual i (in this paper, individual i’s degree is the number of individuals being within a circle that centered at individual i) and U is a control parameter (U is the same for all individuals). This definition is based on the following consideration: Maintaining a social contact is usually costly [57]. We assume for simplicity that an individual pays U cost to maintain a link with one of its neighbors, and the aspiration level for an individual is defined as the total cost for maintaining social links with all its neighbors. Thus it is reasonable to assume the aspiration level to be proportional to the degree of individuals. For U ≤ 0, the minimal payoff of individuals is higher than the aspiration level, individuals do not move (except isolated individuals). For U > b, the aspiration level is higher than the maximal payoff of individuals, all individuals move. 3. Results and discussions Fig. 1 shows the fraction of cooperators ρc as a function of the aspiration level U for different values of the temptation to defect b when the migration velocity v = 0.2. One can find that for a fixed value of b, there exists a moderate aspiration level U, resulting in the highest cooperation level. Fig. 2 depicts the results obtained for a wider range of model parameters (U and b). As shown in Fig. 2, there exists an intermediate aspiration level, resulting in the highest cooperation level for different values of b. Additionally, the maximal cooperation level at the optimal intermediate aspiration level decreases with the increase of b.
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Fig. 1. (Color online) The fraction of cooperators ρc as a function of aspiration level U for different values of the temptation to defect b. The population size N = 1024, the interaction radius r = 1, the migration velocity v = 0.2 and the linear size of square plane L = 25. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
Fig. 2. (Color online) The color code shows ρc as a function of U and b together. Other parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators’ results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
It is known that in spatial games, cooperators can survive by forming clusters [58,59], thus reducing exploitation by defectors. For low aspiration levels, most individuals do not move, thus cooperator clusters and defector clusters keep relatively steady, restraining the pervasion of cooperators among population. On the contrary, for high aspiration levels, most individuals move. Due to the frequent change of neighbors, cooperators cannot form steady clusters to resist invasion of defectors. As a result, cooperators are extinct and defectors dominate the whole population. For moderate aspiration levels, cooperators are able to form steady clusters since the benefits of mutual cooperation are high, whereas defector clusters are unstable because the payoffs of mutual defection are low. From Fig. 3, one can see that for moderate aspiration levels, the number of mobile defectors is much more than that of mobile cooperators during the process of evolution. Furthermore, one can see that the number of mobile cooperators decreases as time evolves, while the number of mobile defectors increases firstly and then decreases as time evolves. Once a mobile defector approaches the boundary of a cooperator cluster, he/she has a large probability to change to a cooperator since cooperator clusters usually obtain a high payoff. Compared with the static case in which cooperator clusters and defector clusters are almost changeless, a moderate aspiration level leads to the expansion of cooperator clusters and the collapse of defector clusters. So, cooperation is promoted when the aspiration level is moderate. To confirm the above analysis, we study the fraction of cooperators ρc as a function of time for different aspiration levels. As shown in Fig. 4, for a low aspiration level (U = 0), ρc reaches a steady value, indicating that cooperator clusters
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Fig. 3. (Color online) The number of mobile cooperators nc and mobile defectors nd as a function of time step t. The population size N = 512, the interaction radius r = 1.3, the migration velocity v = 0.3, the temptation to defect b = 1.1, the linear size of square plane L = 25 and the aspiration level U = 0.5.
Fig. 4. (Color online) The fraction of cooperators ρc as a function of time step t for different values of U. Other model parameters are the same as those in Fig. 3.
and defector clusters coexist and remain stable in the equilibrium state. For a moderate aspiration level (U = 0.5), ρc decreases in the beginning and then increases to unity as time evolves. This is because cooperators are initially dispersive and they are vulnerable to the attacks of defectors, resulting in the decrease of ρc at the beginning. After cooperators gather together to form clusters, cooperator clusters will expand continually when the aspiration level is moderate, thus ρc increases. For a high aspiration level (U = 1.1), ρc decreases to zero, indicating that cooperators are extinct in the end. We provide an intuitionistic observation for the evolution of cooperation in supplementary movies S1–S3 (available online at doi:10.1016/j.physa.2010.07.034). Movie S1, S2 and S3 corresponds to U = 0, 0.5 and 1.1 respectively. In all movies, blue spots represent cooperators and red spots denote defectors. The migration velocity v also affects the evolution of cooperation. Low migration velocity favors cooperation while high migration velocity promotes the spread of defection. As shown in Fig. 5(a), the fraction of cooperators ρc decreases as the migration velocity v increases for different values of the aspiration level U. Fig. 5(b) shows that ρc as a function of U for different values of v . One can see that, for v = 0.005, 0.1 and 0.2, the cooperation level is highest when U is moderate. Additionally, the maximal cooperation level at moderate aspiration level decreases with the increase of v . In particular, for high migration velocity (v = 0.5), cooperators can survive when the aspiration level U = 0 (note that U = 0 corresponds to the static case in which individuals do not move), while defectors dominate the whole system for U > 0. For low migration velocity, a mobile defector slowly approaches the boundary of the cooperator cluster and he/she is likely transfer to cooperator due to the influence of cooperators locating inside cooperator clusters. However, for high migration velocity,
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b 0.005 0.1 0.2 0.5
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Fig. 5. (Color online) (a) The fraction of cooperators ρc as a function of the velocity v for different values of U. (b) ρc as a function of the aspiration level U for different values of the velocity v . The temptation to defect b = 1.25, other model parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps (for v = 0.005, the equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 106 time steps). Each data point is an average over 500 different realizations.
Fig. 6. (Color online) The fraction of cooperators ρc as a function of the interaction radius r for different values of U. The temptation to defect b = 1.2, other model parameters are the same as those in Fig. 1. The equilibrium fraction of cooperators results from averaging over 2000 time steps after a transient period of 105 time steps. Each data point is an average over 500 different realizations.
a mobile defector will rapidly enter the core of acooperator cluster, thus gaining high payoff and resulting in the collapse of the cooperator clusters. The neighborhood size plays an important role in cooperation [60]. Fig. 6 studies the fraction of cooperators ρc as a function of the interaction radius r for different aspiration levels. One can see that there exists a moderate value of r, leading to the highest cooperation level. For a small interaction radius r, individuals have few neighbors and cooperators cannot form clusters to resist the invasion of defectors. For a large interaction radius r, individuals’ neighborhoods resemble a wellmixed population in which more or less everybody interacts with everybody, so that the low cooperation level is inevitable. 4. Conclusion In summary, we have studied the effect of aspiration-induced migration on the evolution of cooperation. Individuals are located on a square plane and gain payoffs by playing the prisoner’s dilemma game with those within a circle of radius r. Individuals will leave their current locations if their payoffs are lower than the aspiration level. We have found that there exists a moderate aspiration level, leading to the highest cooperation level. Additionally, cooperation is enhanced if migrants move with low velocity but inhibited if they move with high velocity. For a moderate aspiration level plus a low migration velocity, cooperator clusters outspread and defector clusters disintegrate, resulting in the prevalence of cooperation amongst the population. Moreover, we have reported that a moderate interaction radius can best promote cooperation. Our study may be of practical significance and help to explain why cooperation is ubiquitous in nature. Furthermore, our work may encourage in-depth studies on the role of migration in evolutionary game theory.
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Acknowledgements We thank Hisashi Ohtsuki for helpful discussions. This work is funded by the National Basic Research Program of China (973 Program No. 2006CB705500), the National Natural Science Foundation of China (Grant Nos. 10975126, 10635040), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20093402110032). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]
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